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Ergodicity and Hopf–Lax–Oleinik formula for fluid flows evolving around a black hole under a random forcing

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Abstract

We study the ergodicity properties of weak solutions to a relativistic generalization of Burgers equation posed on a curved background and, specifically, a Schwarzschild black hole. We investigate the interplay between the dynamics of shocks, a curved geometric background, and a random boundary forcing, and solve three problems of independent interest. First of all, we consider the standard Burgers equation on a half-line and establish a ‘one-force-one-solution’ principle when the random forcing at the boundary is sufficiently “strong” in comparison with the velocity of the solutions at infinity. Secondly, we consider the Burgers–Schwarzschild model and establish a generalization of the Hopf–Lax–Oleinik formula. This novel formula takes the curved geometry into account and allows us to establish the existence of bounded variation solutions. Thirdly, under a random boundary forcing in the vicinity of the horizon of the black hole, we prove the existence of a random global attractor and we again validate the ‘one-force-one-solution’ principle. Finally, we extend our main results to the pressureless Euler system which includes a transport equation satisfied by the integrated fluid density.

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Notes

  1. Clearly, our assumption \(u_0 \in L^1([r_*, +\infty ))\) is equivalent to \(U_0 \in L^\infty ([r_*, +\infty ))\), since the factor \(1-2m/r\) remains bounded near \(r_*\) and tends to 1 at infinity.

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Acknowledgements

The work of YB was partially supported by the National Science Foundation (NSF) through Grant DMS-1460595. This work was done when PLF was a visiting researcher at the Courant Institute of Mathematical Sciences (NYU) and was also partially supported by the Innovative Training Networks (ITN) Grant 642768 (ModCompShock).

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 215 Mercer Street, New York, NY, 10012, USA

    Yuri Bakhtin

  2. Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique, Sorbonne Université, 4 Place Jussieu, 75252, Paris, France

    Philippe G. LeFloch

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  1. Yuri Bakhtin
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  2. Philippe G. LeFloch
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Correspondence to Philippe G. LeFloch.

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Bakhtin, Y., LeFloch, P.G. Ergodicity and Hopf–Lax–Oleinik formula for fluid flows evolving around a black hole under a random forcing. Stoch PDE: Anal Comp 6, 746–785 (2018). https://doi.org/10.1007/s40072-018-0119-8

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  • DOI: https://doi.org/10.1007/s40072-018-0119-8

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